On an inequality for the Smarandache function J. Sandor Baheยง-Bolyai University, 3.JOO Cluj-Napoca. Romania
1. In paper [2J the author proved among others the inequality S(ab) :::; as(b) for all a, b positive integers. This was refined to
S(ab) :::; S(a)
+ S(b)
(1)
in [1 J. Our aim is to shmv that certain results from om recent paper [3] can be obtained in a simpler way from a generalization of relation (1). On the other hand, by the method of Le [lJ we can deduce similar, more complicated inequalities of type (1). 2. By mathematical induction we have from (1) immediately:
(2) for all integers ai 2 1 (i
= L ... , n).
When
al
= ... = an = n
we obtain
(3) For three applications of this inequality. remark that
S((m!t) :::; nS(m!) = nm
(4)
since S( m!) = m. This is inequality 3) part 1. from [3J. By the same way, S( (n!)(n-l)!) :::;
(n - l)!S(n!) = (n - l)!n = n!, i.e.
(5)
125